Doubly-periodic wave filter



June 's, 1926. 1,587 514 Q. J. ZOBEL DOUBLY PERIODIC WAVE FILTER 5/ Filed April 30. 1920 2 Sheets-Sheet l 5/ I jy/gll INVENMR.

@4/ By 0. .1 20M c. ATTORNEY Patented June 8, 1926.

UNITED- STATES PATENT OFFICE.

OTTO J. ZOBEL, OF MAPLEWOOD, NEW JERSEY, ASSIGNOR TO AMERICAN TELEPHONE AND TELEGRAPH COMPANY, A CORPORATION OF NEW YORK.

DOU'BLY-PEBIODIC WAVE FILTER.

Application filed April 30, 1920. SeriaINo. 379,966.

This invention relates to selective circuits, and more particularly to selective circults of the type known as wave filters.

A number of wave filters are described in certain patents issued to George A. Campbell, 1,227,113 and 1,227,114, dated May 22, 1917. The Campbell patents disclose three types of wave filters, namely: Low pass filters, high pass filters and band filters. A low pass filter is one which transmits with negligible attenuation a band of frequencies extending from zero to some preassigned upper limit. A hlgh pass filter is one which transmits with negligible attenuation a band of frequencies extending from a pre-assigned lower limit to infin1ty. band filter, however, transmits a band of frequencies extending over a range whose upper and lower limits both he between zero and infinity.

Five different forms of band filters are described in the Campbell patents, these five forms being illustrated in Figures 1 to 5 inclusive of said patents. The four forms of band filters illustrated in Figs. 2 to 5 inclusive of the Campbell patents are subject to the objection that while the cut-off at one side of the band of free transmiss on is fairly sharp, the cut-off at the other side ofthe band is much more gradual. In the case of the band filter illustrated in Fig. 1 of the Campbell patents, while the cut-off is sharp on both sides of the band, very great precision in the design of the elements of the filter is necessary in order to secure uniform transmission in the range of free transmission. This is due to the fact that this particular type of filter is in reality a filter transmitting two bands wh ch are caused to merge or coalesce by making the upper limit of the lower band and the lower limit of the upper band of the same frequency. If these two frequencies are not made identical by careful design of the elements of the filter, an attenuation region may occur about the middle of the band of free transmission of the filter which, unfortunately, is the portion of the filter that in practice is used the most.

1t is, therefore, one of the objects of this invention to provide a type of wave filter which will transmit a band of frequencies having a very sharp cut-off at each side of the transmitted band, said filter being composed of elements which do not requlre extreme accuracy in their design.

Another object of the invention is to provide a filter Whose structure shall be doubly per od cas distinguished from the singly periodic structures of the filters disclosed in the Campbell patents above referred to. Other and further objects of the invention Wlll appear more fully hereinafter.

- The objects of this invention are attained by constructing each section of the doubly periodic filter of two parts, one of which may be a section of one of the forms of filters disclosed in Figs. 2 to 5 of the Campbell patents, and the other part of which may be a section of another of the several types of filters disclosedin said figures of the Campbell patents. An analysis of these figures of the Campbell patents shows that there are two filters, each having the same type of series element in each section, but having different shunt elements. In one form of the. filter, in accordance with the present invention, each section will comprise a sectionfrom each of these two types of filters, so that the section of the doubly periodic structure will consist of two similar series elements with different shunt elements. The other two types of filters in the Campbell patents, it will be noted, each have. the same type of shunt element, but

different series elements. Another type of filter in accordance with the present invention, may, therefore, be obtained by combining in each of its sections a section from each of these two types of filters, so that the resultant section will have two similar shunt elements, but different series elements.

Of these two types of doubly periodic filters, one has a characteristic impedance at a mid-series termination of a half section of the filter, and the other a characteristic inipedance at a mid-shunt termination of a half section of the filter, which in either case is equivalent to the characteristic. impedance of the correspondingly terminated Campbell filter of the type illustrated in Fig. 1 of the Campbell patents. For convenience, the first of the above described types of doubly periodic. filters is called the mid-series doubly periodic single band wave filter. This filter is so called from the fact that it is produced by connecting alternately in series single mid-series sections of two different. 3-element single band wave filters whose mid-series characteristic impedances are equal to that of the sin le band type illustrated in Fig. 1 of the ampbell patents. The series element in both of these component sections is an inductance in series with a capacity, while the shunt element is in one a capacity and in the other an inductance.

In the other type herein called the midshunt doubly periodic single band wave filter. the double periodic section 1s produced by alternately connecting in series single mid-shunt sections of two different 3-element single band wave filters whose mid-shunt characteristic impedances are equal to the corresponding impedance of the single band filter illustrated in Fig. 1 of the Campbell pa-tents. The shunt element in both these two component sections is an inductance in parallel with a capacity, while the series element is in one a capacitv and in the other an inductance.

The term mid-series termination as used in this description has reference to such a termination of the filter that the end section shall terminate in a series element whose impedance is half that of the normal series element. Similarly, the term mid shunt termination means such a termination of a wave filter that the end section of the series shall terminate in a shunt element whose admittance is one-half of the admittance of the normal shunt element. In other words, the impedance of the terminal shunt element is twice that of the normal shunt element.

From these definitions it follows, of course, that the mid-series impedance is the impedance measured at a mid-series termination, while the mid-shunt impedance is the impedance measured at a mid-shunt termination.

The invention may now be more fully understood from the following description, when read in connection with the accompanying drawing in which 2- Fig. 1 illustrates a confluent band filter of the type illustrated in Fig. 1 of the Campbell patents above referred to.

Fig. 2 illustrates a filter having the characteristics of the filter of Fig. 5 of the Campbell patents.

Fig. 3 illustrates a filter corresponding to the filter of Fig. 3 of the Campbell patent.

Fig. 4 illustrates a filter similar to that shown in Fig. 2 of the Campbell patent.

Fig. 5 illustrates a filter corresponding to that of Fig. 4 of the Campbell patent.

Figs. 6 to 10 inclusive illustrate how sections of the filters of Figs. 2 and 3 may be transformed into mid-series terminations and combined to form a single section of a doubly periodic filter.

Fig. 11 is a diagram illustrating a plurality of sections of the resultant mid-series doubly periodic single band filter.

Figs. 12 to 16 inclusive are diagrams showing how sections of the filters of Figs. 4 and 5 may be transformed into mid-shunt terminated sections and combined to form one section of a doubly periodic filter.

Fig. 17 is a diagram illustrating a number of sections of the resultant mid-shunt doubly periodic single band filter.

Figs. 18 to 23 inclusive are schematic diagrams used in connection with the development of the mathematical theory underlying the wave filter of this invention.

The mid-series double periodic type will be considered first, and in order to understand the theory underlying this filter it will first be necessary to develop the design formula of the confluent band type of filter illustrated in Fig. 1.

Pre-determtnatz'on of intimate/noes and capam'tz'es in the confluent band constant is wave filter.

. The confluent band filter illustrated in Fig. 1 has a characteristic such that the product of the impedances of its series and shunt elements is e ual to the square of a constant, called for convenience, k. Hence, the term constant k filter. The filters illustrated in Figs. 2 to 5 inclusive do not have this characteristic, although doubly periodic filters are herein developed from these types which will have a characteristic impedance equal to that of the constant 7c filter just referred to.

While design formulae are given in the Campbell patents above referred to for fil ters of the type illustrated in Figure 1, it will be more convenient for the purpose of this specification to derive these design formulae in terms of the limiting frequencies and the constant is.

It is well known that a smooth line having uniform series impedance distributions 2,, and uniform shunt impedance distributions 2 per unit length has the character istic impedance and a propagation constant A wave filter such as illustrated in Fig. 1, having series elements 2 and shunt elements 2 has a characteristic impedance at an; termination which is a function of both the product. and the ratio of a, and 2 and has a propagation constant which is a function of their ratio, hence it has been found convenient to express both the characteristic impedance and propagation constant of the wave lillcr in terms of It and y, the parameters of the corrcs minling smooth line. As previously stated. the filter of Fig. 1 is of the constant It type, and in this type of lilter it is necessary that the product of the series impedance 2, and the shunt impedance 2 shall be a constant independent of frequency, that is,

As pointed out in the above mentioned patents to Campbell, free transmission takes place through such a network for frequencics which give within the range whose limits are defined by 2 -2 k a constant.

By combining equation 5 with equation 3, it is apparent that the filter trans mts within a range of frequencies in which 2 hes between In Fig. 1, the series element .2 consists of an inductance L in series with a capacity 0,, and the shunt element 2, consists of an inductance L in parallel Wlth a capacity C. Hence 2 0 and 2 i'i2lc we have, by substituting these values in equation 7 By substituting these two values of .2 in equation 11, we get The solution of equations 12 gives Substituting these values of a and b in equations 9 and 10, We have as the design formulae for the series inductance and capacity 1 Llzmll (14) (f2"f1) Z Similarly, the product of the impedance of the capacity C, and the impedance of the inductance L will be equal to Hence,

Solving these equations We at once have the design formulae for the shuntinductance L and shunt capacity 0 as follows:

- tL 21r f= k (16) Prcdetermination 0f the constants of the single band filters of Figures 2 and 3.

The desired mid-series doubly periodic type of filter in accordance with this invention is constructed by making up each complete section of alternate mid-series sections of the filters of Figs. 2 and 3. As shown, the impedance .2, in Fig. 2 comprises an inductance L 'and capacity C, in series, While the shunt impedance 2 consists of a capacity C Likewise, in Fig. 3, the series element 2 of each section conslsts of an inductance L and capacity C," in series, While the shunt element .2," consists of an-inductance istic impedance ofeach of these types of filters pedances z,-

must be equal to that of the constant 70 type illustrated in Fig. 1. .The manner in which alternate sections are to be combined to form the doubly periodic type is illustrated in Figs. 6 to 10 inclusive. Figs. 6 and 9 illus\ trate single sections of the filters of Figs. 2 and 3 respectively. As indicated in Fig. 7, a filter section equivalent to that of Fig. 6 may be constructed by connecting the shunt impedance 2,, at the mid-point of a series impedance made up of two series elements each having a. value equal to half of the im- It will be seen that if a filter of the ty e illustrated in Fig. 2 be made up of a number of sections such as illustrated in Fig. 7, the filter will obviously terminate in mid-series, since the terminal element of the filter will have an impedance equal to onehalf of the full series element.

In a similar manner the filter section of Fig. 9 may be transformed into-an equivalentstructure illustrated in Fig. 8 by connecting the shunt impedances 2, at the midpo1nt of a series impedance made up of-two elements each having one-half the impedance of a full series element. If, now, the two sections illustrated in Figs. 7 and 8 have the same mid-series characteristic impedances, there will be no reflection losses if the sections be connected in series as illustrated in Fig. 10, in which the adjacent elements 1 1 2 .2 and 2 are combined to form the series impedance element The network of Fig. 10 comprises a complete section of a mid-series doubly periodic filter. By combining a plurality of such sections, a complete wave filter of the midseries doubly periodic type will be constructed as illustrated in Fig. 11. In this The above relation is sufficient to obtain all of the inductances and capacities of the filters of Figs. 2 and 3 in terms of f,, f, and k. The values of these inductances and capacities will now be obtained.

figure the relations of the various inductances and capacities to the corresponding inductances and capacities of Figs. 2 and 3 are illustrated. It will be noted that the filter terminates at the left in a mid-series section whose series element has a value equal to one-half of the full series element of Fig. 2. Similarly the filter of Fig. 11 terminates at its right hand end in a mid-series section whose series element has one-half the value of a full series element of Fi 3.

Let us now' determine the mid-series characteristic impedance Z of the constant [a type of wave filter illustrated in Fig. 1. A Wave filter of this type is schematically indicated in Fig. 18. From this figure it is apparent that Also, from this figure we may express the characteristic impedance Z measured across the terminals of a full section, as,

which may be written Solving equations 18 and 19, we have as the expression of the mid-series impedance of the filter of Fig. 1

The mid-series impedances of the types of filters illustrated in Figs. 2 and 3 may obviously be determined in a similar manner and may be expressed as follows:

Figs. 2 and 3 are to be made equal to that of the filter of Fig. 1, we have ms ms ms and hence 2.1/2 all/ 21! i lllg For the constant is type of filter illustrated in Fig. 1, We have Substituting in equation 25 the value of a,

as given by' equation 7, we have 1 .1-LO4 z 2 2 (Z =k (2 By analogy to equations 7 and 8 we have as the values of the series element 2, and the shunt element 2 of the filter of Fig. 2 the following:

Substituting these values in equation 28, We

have

1 1 1 1 1 1 We? W) F (31 So, also, for the filter of Fig. 3 we have Again, by analogy to equations 7 and 8 we have I (201/ C2) A L1 zo 'o ho.

Equation 36 in connection with equation 14 gives the following as the design formula of the filter of F1g. 2:

It will be observed that the above design formulae are obtained from the design formulae of the constant k filter of Fig. 1.

In a similar manner the design formulae for the filter of Fig. 3 may be obtained by setting the right-hand termof equation 35 Substituting these values in equation 32 we get equal to the right-hand term of equation 27. Again equatlng co-eflicients we have i I/ 1 II 1 L1 0." 2 0* 0 1/ 01 1. 2" 1 Combining the above equations With equation 14, we have as the design formulae for the filter of Fig. 3 the following:

Mid-sem'es doubly periodic. single band. wa've,

filter.

If a mid-series doubly periodic Wave filter is made up of alternating mid-series sections of the two types of wave filters illustrated in Figs. 2 and 3, the values of the inductances and capacities as indicated in Fig. 11 may be at once obtained from the design formulae 37 and 39 of the filters of Figs. 2 and 3. Also, since there will be no reflection losses at the tion points of the components.

It now remains to be shown that the propjuncschematica agation constant of the doubly periodic filter thus constructed will be the same as the propagation constant of the filter of Fig. 1. Referrin to Fig. 21 which illustrates Fly a mid-series doubly periodic wave filter, it is apparent from the relations between applied voltage, current and mpedance that the current in the nth section of the filter flows through an impedance equivalent to the impedance of a shunt element 2, in parallel with the impedance Z of an infinite number of sections. Hence,

V ZIZ I I; Z! +22! where I designates the current in th e nth component section. of the filter. Similarly,

from Fig. 21, since the current in the next half section flows through an impedance Z, we have where I,,,. represents the current flowing in the (n+1)st component section of the filter. From Fig. 21 it is also apparent that the impedance Z is equivalent to an impedance 2, in series with the mid-series impedance But sincefrom equation 23, Z is equal to Z... we have iii i Z 2 +z In (42 which may be written Z1; Z1221 ms ""fi Z! Whe'nce ZZB f a! =Y Combining equations 42 and 44, we have where 6 denotes the base of Naperian logarithms and I" denotes the propagation conwhere we have already chosen the constants of the impedance elements Z and Z so as to make Z =Z :Z as indicated by design formulae 14, 17, 37 and 39.

stant of a wave filter of the type illustrated in Fig. 21. The relation between the first and last members of equation 45 is established by equation 1 of the Campbell patent above referredto.

In a manner similar to the derivation of equation 45 the following relation may also be obtained from Fig. 21:

where I" is the propagation constant of a filter of the type illustrated in Fig. 3.

In an analogous manner the following equation may be derived from Fig. 20.

f =e-1" cI"=-e" where P, designates the ropagation constant of the mid-series dou ly periodic wave filter. Since this propagation constant is to be the same as that of the constant is type of filter illustrated in Fig. 1, we have This is the relation to be proven, and in order to show that this relation holds, it is merely necessary to show that the equivalent relation obtained from equation 45, 46 and 47 holds, namely:

Substituting Z for Z and Z,,,,. in equation 50, clearing of fractions and sunplifyin we have In order to prove equation 51, it is necessary to determine the relationship existing between 2 2 and z From equations 29 and 37 we have and from equations 33 and 39 We have From equations 7 and 14 we get (fr-f1)! (52) away Substituting the value of z, as given in equation 52 in equation 55, we h (ix-fee Also, from equations 52, 53 and 54 the expression 2 +z -.2 may be written as follows Also, from these same equations, the expression 2 .2 .2 may be written I 1/ j :l 2 2 z f1f 2121 21 (flfz f (f1 f flfz f2 Substituting the values given by equations 56, 57 and 58 in equation 52, and cancelling the factor which occurs in both terms, we have, by expansion It will be noted that the terms in equation 59 all cancel out, which proves the relation established by equation 50. Hence the propagation constant of the mid-series doubly periodic wave filter of this invention 18 the same as the propagation constant of the confluent band filter of Fig. 1.

Precleterminatz'on of inducfances and capacities in mid-shunt doubly periodic single band wave filter.

This type of wave filter is illustrated in Fig. 17 and is produced by combining alternately in series sections of the filters of Fig. 4 and Fig. 5 respectively. The manner in which these sections are combined is illustrated in Figs.12 to 16 inclusive. Fig. 12 indicates a single element of the filter of Fig. 4, While Fig. 15 illustrates a single element of the filter of Fig. 5. By dividing the shunt element 2 of Fig. 12 into two similar impedance elements, having twice the value of 2 and arranged in parallel, the equivalent mid-shunt structure of Fig. 13 may be attained. In a similar manner the structure illustrated in Fig. 14 is equivalent to the filter structure illustrated in Fig. 15. By joining together the structures of Figs. 13 and 14, and combining the adjacent shunt elements 22 and 22 as one unit we have the network shown in Fig. 16, which illustrates one section of the mid-shunt doubly periodic filter.

Fi 17 illustrates a filter of this type contpose of a luralit of sections terminating at the ends in a mi -shunt termination. The left-hand termination is a mid-shunt termination for a filter of the type of Fig. 4, while the right end of the filter has a. mid-shunt in parallel with the admittance A measured pcross a full section of the filter. Hence, we iave tance. Also, since the admittance A is the reciprocal of the impedance Z of a full sec- Eli Eli

till

1 termination corresponding to the filter of tional termination We have from 19 Fig. 5. The values of the various induc- 1 tances and capacities in the filter of Fig. 17 1 z Z W (61) are given in terms of the corresponding in- A a-P A 'l' ductances 51nd czpaclilties of the filt7ers 0E 22 z 1 Fi e. 4 an 5. s s own in Fig. 1 eac full section of the filter comprises two half substltltmg A for Z equatlon 60 may be sections having shunt elements of inducexpressed tances and capacity in parallel and series 1 elements which are for the first half sec- A+ ==\/i (i (62) tion a caplacity, and the next half sec- 222 1 2 222 mu tion an in uctance. e series capacity will be the same as that of Fig. 4, and the series 2 5 3 g ig i g the inductance the same as that of F ig. 5. The i 6 f 3 e 'g a We inductance of the shunt element will have a V r equa Ion value of 2L L I 1 zlzz w qgy 1 2 2 and the capacity of the shunt element will have avalue glare Z denotes the mid-shunt imped- 1 In a similar manner it ma be shown that 2' 02") for the ffilter;1 ofl Fig. 4 the mid-shunt admittance o sai fi ter ma be In order that the two half sections may lo y expressed as fol be combined to give a full section of the Z I filter of Fig. 17 each of these half sections Z which are in fact full sections of the filter mflh r l 2 of1 Figs. 4 and 5 anust have a mid-shunt 4 c aracteristic impe ance equal to that of 40 the constant 70 type illustrated in Fig. 1. gfi fiy g t g f gg g f Let us now determine the mid-shunt chary p esse acteristic impedance of the filter of Fig. 1. 2/2 Fig. 19 illustrates schematically a filter of Z 5 this type terminating in mid-shunt. From /z "z (z the diagram of Fig. 19 it is apparent that the mid-shunaadmlttance A measured As the mid-shunt admittance expressed across the termlnals of the filter is equal to in equations 63, 64: and. 65 must be equal,

the admittance of the mid-shunt element 22 we have from these three equations msh 1 I 1 1 1 2 2 4.2 2 z 42 2 z 42 1oo.If, now in equation 63 we substitute k for 2 2 and for 2 substitute its value as given in equation 7, we have for the constant is type of filter illustrated in Fig. 1

1 msh L1 1 1r L1 'f (67) k 20 W. 1611' C k -f k In the case of the filter of Fig. 4 the series and shunt impedance may be expressed as follows:

' mah Likewise in the case of as follows:

thefilter of Fig. 5 the series and shunt impedances may be expressed Substituting these Values in equation 65, We have i I 9 ll L1 from which, in connection with equations 14, we get the expressions for theinductances and capacities of the filter of Fig. 4 as follows:

Similarly, by equating the co-eflicients of 67 and 71 we get 1 I 1 1 L Uzi/(LT,

from which, in connection with equations 14, the expressions for the inductances and The inductances and capacities for the filter of Fig. 17 may .at once be obtained from the values given by formulae 73 and 75. I g

It now remains to be shown that the propagation constant of the filter of Fig. 17 is the same as that of the filter of Fig. 1. Referring to Fig. 23, which is a schematic diagram of a filter of the type of that of Fig. 17, it will be seen that a complete section of the doubly periodic filter may be considered. as being made up by directly connecting together the elements of Figs. 12 and 15.

From Fig. 23 it will be noted that the current I flowing into the mid-shunt impedance Z of the doubly periodic filter divides, part flowing through the impedance 22,, and the remainder,

n+5 through the parallel full series impedance, Z. Since for the same voltage drop across Z and Z the currents through them are inversely as their impedances, we have It is also evident from Fig. 23 that the midshunt impedance Z,,.,,. is equivalent to the impedance 22 in parallel with the impedance Z, hence,

z z z This equation may be rewritten;

Combining equations 76 and 78 we obtain 1 msh In a similar manner from Fig. 23

1 From equations 79 and 80 and since Z Z we have, as the expression for the ratio of currents in the nth and (n+1)st half section of the doubly periodic filter the following In the same manner it may be shown from Fig. 23 that the expression for the ratio between currents in the (n+2) ndand (n+1)st half sections is n+1 2 msh n 2+ msh Now, the ratio between the currents flowing in two adjacent full sections of the filter of Fig. 23 is the ratio of the current I to From equations 89 it is apparent that the current I Hence, from equations 81 and 82 we have where 1 is the propagation constant of the.

doubly periodic filter.

Now, in order that the doubly periodic filter of Fig. 17 be equivalent in its transmitting properties to the coalescent band filter of Fig. 1, it is necessary that both filters have the same propagation constant; hence the following relation must hold true.

In order to show that the relation of equa tion 85 holds, it is only necessary to prove that the following equivalent relation holds:

222 msh 2Z2 msh 222 I, msh 2Z2 msh n 222, msh 222 I, msh

Since msh msh'= mh" this reduces to msh 'l 2 2 2 2 2 2 :0 It is now necessary to determine the relation between the impedances-.2 .2 and'z These impedances may be expressed as follows:

Now, since in a constant 70 filter the product of the series and shunt impedances z,

- .2 must be equal to 70 it follows that the products of the mid-shunt impedance and mid-series impedance (which in a singly periodic filter always equals the product 2, 2 must also be equal to 70 From this relation, taken in connection with equation 56, we have Substituting the values common factor, we get given by equations 90 and 91 in equation 88, and omitting the Clearing of fractions and expanding this equation reduces to It will be seen that all of the terms in equation 93 cancel out, which proves the assumed relation of equation 85. Consequently the propagation constant of the mid-shunt doubly periodic wave filter is the same as that of the constant 70 type of Fig. 1.

From the above mathematical exposition of the properties of the filters of- Figs. 11 and 17, it is apparent that either of these doubly periodic filters may be designed to have the same critical frequencies, and the same sharp cut-off as the constant k filter of Fig. 1. Furthermore, since these filters are inherently single band filters, their elements need not be so carefully proportioned in order to avoid attenuations at the middle of the transmitted band, thereby avoiding the difliculty occurring in the filter of Fig. 1 where the single band is formed by causing two bands to coalesce.

It will be obvious that the general principles herein disclosed may be embodied in many other organizations widely different from those illustrated, without departing from the spirit of the invention as defined in the following claims.

What is claimed is 1. A selective circuit for selectively transmitting a band of frequencies comprising a plurality of similar sections, each section consisting of two part sections and each part section being made up of two impedance elements, one in series and one in shunt, one of said impedances being a combeing a capacity in the case of one part section and an inductance in the case of the other part section, the said part sections of each kind bein so designed that a complete selective clrcuit of only one such kind will have the same critical frequencies as a complete selective circuit of the other kind.

2. A selective circuit for selectively transmitting a band of frequencies comprising a plurality of similar sections, each section consisting of two part sections, said part sections being made up of two impedance elements, one in series and one in shunt, one of said impedances being the same for both part sections and the other impedance element being in the one case a capacity and in the other case an inductance, the said part sections of each kind being so designed that a complete selective circuit of only one such kind will have the same critical frequencies is :5 complete selective circuit of the other 3. A selective circuit for selectively transmitting a band of frequencies comprising a plurality of elements connected successively in series, each element consisting of two impedances, one in series and one in shunt, one of said impedances being the same for each element, the other impedance being of two types arranged alternately with regard to successive elements, the said impedances being so proportioned that a selective circuit made up Wholly of alternate midseries or midshunt sections will have the same critical frequency.

4. A selective circuit for. selectively transmitting a band of frequencies comprising a plurality of elements, each element including a pair of impedances, one in series and one in shunt, one of said impedances being composed of a similar combination of inductance and capacity for each element, the other impedance being a capacity in the case of every other element, and an inductance in the case of the remaining elements, the said impedances being so proportioned that a selective circuit made up wholly of alternate midseries or midshunt sections will have the same critical frequency.

5. A composite single transmission band wave filter comprising alternately arranged part sections taken from two other sin le transmission band filters having perio ically recurring sections and having sharp cut-off respect1vel at op osite ends of the transmission ban where y the composite filter has sharp cut-off at both ends of its transmission band.

6. A composite single transmission band filter having alternate elements drawn respectively from each of two di-fierent simple single transmission band filters giving sharp cut-ofl respectivel at opposite ends of the transmission ban whereby the composite filter is given a sharp cut ofi at both ends.

7. A composite single transmission band filter havin the same characteristic impedance an the same ropa ation constant as a certain confluent and lter, and comgrising alternately disposed part sections rawn respectively from two single band filters of the same characteristic impedance and with sharp cut-off respectively at opposite ends of their transmission bands, whereby the composite filter is given a sharp cutoff at both ends thereof.

8. A composite. single transmission band wave filter comprising two kinds of part sections alternatel arranged, the part sections of one kin being adapted in combination to give a sharp cut-off at one end of the transmission band and the part sections of the other kind being ada ted in combination to give a shar cut-o at the op osite end of the transmission band.

11 testimony whereof, I have signed my name to this specification this 28th day of April 1920.

' OTTO J. ZOBEL. 

